MATHEMAICAL MODEL FOR COMBINED OUTBOUND AND

INBOUNDED LOGISTICS USING MILKRUN DELIVERIES

Thanatorn Sinphatsirikul

Pathorn Nantaseri

Sukachai Penpokai

Korapat Yusatharporn

Aussadavut Dumrongsiri† Sirindhorn International Institute of Technology, Thammasat University

Pathum Thani, 12000, THAILAND

Tel. +662-501-3505, Email: aussadavut@siit.tu.ac.th

Abstract We study a firm managing the inbound and outbound logistics using milk-run

deliveries. We develop a mixed integer linear programming model to plan the routing of

multiple trucks to pick up shipments from suppliers and deliver shipment to customers.

Unlike other researches, our model allows planning combined inbound and outbound

milk-run deliveries. The routing allows the trucks to visit suppliers and customers

without any restrictions on the sequence. As the number of locations to visit is larger

because of the combined location, the trucks can shorten their routes without coming

back to the distribution center. We conduct a numerical experiment. The combined

outbound and inbound milk-run delivery can generate total cost saving of 38.6% and

87.1% for high demand and low demand case respectively. The total distance and

average travel time are also improved at the minimum of 42.5% and 26.4% respectively.

Lastly, the truck capacity utilization is improved 79.9% and 107.1% for high demand and

low demand cases respectively. The observed saving and improvement is significant

showing it as viable logistic cost reduction method. The saving and improvement tend to

be higher for the case with low demand.

1. INTRODUCTION Many firms are forced to perform the transportation analysis to find and

implement the best optimum solution of minimizing the cost of transportation and time.

A reduction of transportation cost and time results in an increase of the whole

organization profitability. Logistics plays a vital role in economic systems. The cost of

logistics operations accounts a large proportion of the value of general commodities in

complex supply chains and globalization. However, innovation and technological

development allows new cost cutting opportunities. (Dianwei, 2006).

Many researchers and institutions have been focused and analyzed the statistics of

international logistics cost. The Council of Supply Chain Management Professionals

(CSCMP) presented that there was an increased in USA logistics costs from 7.8% of

GDP (USD 1,100 billion) in 2009 to 8.3% of GDP (USD 1,211 billion) in 2010. The

Office of the National Economic and Social development Board (NESDB) reported that

logistics cost of Thailand in 2008 was equivalent to 18.6% of GDP (1.7 trillion baht).

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The milk-run logistics method can offer viable solution to the problem. Milk run

delivery allows a truck to deliver products from a single originating location to multiple

receiving locations or pick up products from multiple originating locations to a single

receiving location. Scheduling milk run deliveries is a much more complex task than

scheduling direct deliveries. Decisions must be made about deliveries quantities of

different products, about the frequency of deliveries, and most importantly about the

routing and sequencing of pickups and deliveries (Hugos, 2006). Due to consolidation of

small shipments, milk-run system results in reduction in cost of transportation, travelling

path and fuel consumption. In addition, milk run method is used to improve loading rates

at possible levels and reduce the number of trucks and travel distances. As a result, it is

an effective cost reduction technique.

Empty backhaul is one of the major factors that raise the logistics cost in most

companies. According to a study of Council of Engineers in year 2006, Thailand freight

movement had 690,000 trucks with more than 71.74 million trips and total distance for

truck run was around 12,415 million kilometers per year. 46% of the figure was empty

truck runs, or 33 million trips that included 5,586 million kilometers per year (Peetijade

and Bangviwat, 2012). This problem can also be reduced by combined inbound and

outbound milk run logistics, resulting in lower vacant space in the return trip. In order to

eliminate empty backhaul, we develop a mathematical model to provide the optimal

routes of delivery and pickup route for multiple trucks. The route consists of both

delivery and pickup sequences combined in one truck. Different from other researches,

our model allows the mixture of pickup and delivery sequences to lower the

transportation cost.

Many researchers study cost-reductions using milk-run logistics. Raju, Ajitkumar,

and Dhake (2007) stated that implementation of the milk run system had substantially

reduced excess inventories in the stores, transportation cost and inventory carrying cost.

Also, turning attention to inbound logistics will not only reduce costs, but also generate

more collaborative relationship with internal organizations, suppliers and logistic

providers, which will ultimately result in a more effective business process from which

all parties will benefit. Yildiz et al. (2010) developed a network flow model combined

with vehicle routing from the cross dock and the plant to minimize the system-wise

logistics costs. They identified the opportunity for significant cost savings by using a

mixed-integer programming model that matches opposite flows from and to the

customers and suppliers. Sadjadi et al. (2008) suggested the advantage of running milk

run logistics system that the performance of the supply chain and the logistic is improved

due to effectively using of the transporting vehicles’ spaces, controlling the transport

charges, as well as reducing the level of parts inventory and their maintenance costs.

In this paper, we develop a mathematical model allowing a firm to plan routes of

multiple trucks picking up and delivering the suppliers and customers at the same time.

Due to complexity of keeping records of inventory on the truck during the delivery and

pickup, most researchers limit the delivery to be completed first before pickup can start.

Unlike other researches, our model allows a truck to deliver a shipment at a customer,

pick up a shipment from suppliers and then deliver a shipment at another customer,

without the restriction on sequences, to lower logistics cost. As a number of locations to

visit larger due to combined pickup and delivery locations, the company can plan the

routes more effectively to lower transportation cost and reduce empty backhaul. The

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model is tested in numerical experiments to show the benefits and cost-saving from using

the method. The heuristics algorithm will be developed in the future research.

2. MODEL In our model, a firm manages the inbound and outbound logistics. The firm owns

multiple trucks and needs to pick up shipments from multiple suppliers and deliver

shipments to multiple customers. Each truck could have different capacity to carry load.

Each truck starts its route at a distribution center and end at the distribution center. A set

of retailers and a set of suppliers request deliveries and pick-up, respectively. A truck can

visit customers and suppliers to deliver shipments or pick up shipments. There is no

restriction on sequence of a truck routing pattern. A truck has limited capacity which

could be different for each truck. The amount of load on truck must not exceed the

capacity. Multiple trucks can help deliver or pick up shipments at a node to fully satisfy

the delivery or pick-up demands at the node. A truck has time limit to return to the

distribution center. The shipments to be delivered to customers are loaded onto a truck

when it leaves the distribution center. The company incurs the fixed cost of using each

truck and variable costs to travel between any two nodes. The objective of the firm is to

minimize the total cost consisting on the fixed cost and variable costs. We formulate a

mixed integer linear programming model to solve the problems. The following are the

details of the model.

Parameters

1 .

kij

kij

j

j

Let node is a distributioncenter where all thetrucks arelocated initially

c Cost to transport one unit of shipment from i to j using truck k

T Travel time from i to j using truck k

D Delivery required at Node j

P Pick up required

k

k

at Node j

N Maximum number of nodes

M Total number of trucks

C Capacity of truck k

ct Fixed cost of using truck k

PT Maximumtransportationtime allowed

L Maximum number of loops a truck is allowed

Sets

1

2

{2,..., }

{0,1,2,..., }

{0,1,2,..., 1}

{1,2,..., }

{1,2,..., }

{1,2,..., }

DN Set of demand nodes N

T N

T N

T N

K M

I N

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Decision variables

,

1 ; 0

1 ; 0

kijt

k

kjt

kjt

k t

x if truck k travels from node i to j and arrive node j in period t otherwise

y if truck k is used otherwise

d delivered quantity in period t at node j by truck k

p pick up quantity in period t at node j by truck k

l Inventor

y level of load ontruck k at theend of period t

2

( 1) 1

1

1 1

(1)

1

.

.

., , (2)

, , , (3)

, .

, .

(4

.

)

kij kijt k k

kijt

kilt

k K t T i I j I k K

i I j I

i I j I

j I

j

klj t

k jt k

t T

k jt ki t

t T tD i DNTN

Minimize Total Cost c x y ct

Subject to

x k K t T

x x k K t T l I

x y L k K

x x k K

2

2

2

, , 1

,

1

0

, 2

2

(5)

, (6)

, (7)

, , (8)

, (9)

, , (10)

, , ,

.

.

(11)

,

.

.

.

.

,

kjt

kjt

k t k t kjt kjt

k kjt

k t k

kjt kijt k

j

t T k K

j

k

t T k K

j I

t T j

jt kijt

i

k

I

i I

d D j I

p P j I

l l p d k K t T

l d k K

l C k K t T

d x C k K j I t T

p x C k K j

2

110

, (12)

, (13)

1, (14

.

. )

.t T i I

kijt ijk

k

j I

I t T

x T PT k K

x k K

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Equation (1) represents the total cost which is the objective function of the model.

The total cost comprises the total variable cost and total fixed truck cost. Equation (2)

states that a truck can be used once at time t. Equation (3) equates the number of in-flow

truck and out-flow truck at any node. Equation (4) sets the relationship between the truck

usage variables and the truck routing variables. If a truck leaves a distribution center, the

truck usage variable is set to 1. Equation (5) states that if truck k leaves the distribution

center, it must return to the distribution center. Equation (6) sets the number of deliveries

by any trucks at any time to be equal the required delivery at the node. Equation (7) sets

the number of pick-up quantities by any trucks at any time to be equal the required pick-

up quantity at the node. Equation (8) sets up the relationship of inventory of shipment on

a truck at any time. Equation (9) determines the initial load on the truck when it leaves

the distribution center. Equation (10) limits the inventory on the truck to be fewer than

the truck capacity. Equations (11) and (12) state that if a truck deliver or pick up

shipment at a node j and period t, it must visit the node j at period t. Equation (13) limits

the travel time of a truck to be less than the maximum time allowed. Equation (14) starts

all trucks at the distribution center at period 0 initially.

3. NUMERICAL EXPERIMENT In this section, the model is tested using a numerical example to illustrate the

benefit of using combined inbound and outbound milk-run deliveries. In this experiment,

there are 3 trucks (small, medium and large sizes) to deliver to 4 retailers and pick up

from 3 suppliers. The truck must return to a distribution center within 30 hours. The

parameters of each truck are shown in Table 1. The cost of fuel is approximated at 30

baht per liter. Table 2 shows the distances between all 8 nodes. Table 3 shows the

delivery and pick-up requirements for two scenarios. The first scenario is high demand

for both delivery and pick-up and the second scenario is low demand for both.

Table 1: Information that use for implementation

Table 2: Distance matrix between nodes (kilometers)

Node 1 2 3 4 5 6 7 8

1 0 192 149 97 92 169 137 167

2 192 0 27 107 156 191 86 28

3 149 27 0 166 118 42 150 182

4 97 107 166 0 126 48 36 188

5 92 156 118 126 0 134 109 107

6 169 191 42 48 134 0 57 54

7 137 86 150 36 109 57 0 64

8 167 28 182 188 107 54 64 0

Types of Truck Small Medium Large

Truck Capacity 3.5 tons 6 tons 10 tons

Fuel Consumption 10 km. per liter 6 km. per liter 3.5 km. per liter

Truck Fixed Cost 1500 baht 2000 baht 2250 baht

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Table 3: Delivery and pick up quantities for two cases: High Demand and Low Demand

We solve the mathematical model using IBM ILOG CPLEX version 12.4 using

the data in table 1-3. Figure 1 shows the routings of truck for low demand case when

inbound and outbound milk-run deliveries are planned independently. Figure 2 shows the

routings of truck for low demand case when combined inbound and outbound milk-run

deliveries are planned.

Figure 1: The routings of a truck for pick-up and delivery milk-run deliveries when

planned independently for the low demand case

High Demand Low Demand

Node Delivery Pickup Delivery Pickup

1 0 0 0 0

2 4 0 0.2 0

3 0 4 0 0.2

4 4 0 0.2 0

5 0 4 0 0.2

6 4 0 0.2 0

7 0 4 0 0.2

8 4 0 0.2 0

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Figure 2: The routings of a truck for combined pick-up and delivery milk-run deliveries

for the low demand case

Table 4 and 5 show the results of experiments. We would like to highlight the

benefit of using the combined outbound and inbound milk-run delivery proposed by the

model. We conduct an experiment for the case when the outbound and inbound milk-run

delivery is scheduled optimally but independently. That means a truck must either deliver

or pick up shipment but not both at the same time. Then, we repeat the experiment for the

case but using combined outbound and inbound milk-run delivery which allows a truck to

be able to pick up and deliver in its routing. Table 4 shows the comparison of costs, travel

distances, travel time and truck capacity utilization, defined as the average amount of

load on the truck calculated as average percentage of full truck capacity. Table 5 shows

the percent improvement if the combined outbound and inbound milk-run delivery is

used. The combined outbound and inbound milk-run delivery can generate total cost

saving of 38.6% and 87.1% for high demand and low demand case respectively. The total

distance and average travel time are also improved at the minimum of 42.5% and 26.4%

respectively. Lastly, the truck capacity utilization is improved 79.9% and 107.1% for

high demand and low demand cases respectively. The observed saving and improvement

is significant showing it as viable logistic cost reduction method. The saving and

improvement tend to be higher for the case with low demand.

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Table 4: Results between independent inbound and outbound delivery and combined

inbound and outbound milk-run delivery

Total cost Total

distance

(baht) (km) Truck1 Truck2 Truck3 Average Truck1 Truck2 Truck3 Average

Outbound 10,962.00 850.94 3.23 4.83 6.18 4.75 11.11 42.59 44.44 32.71

Inbound 8,090.57 621.63 3.07 0 7.37 3.48 76.19 0 64.2 46.80

Total 19,052.57 1,472.57 6.3 4.83 13.55 8.23 43.65 21.3 54.32 39.76

Combined

Outbound and

Inbound Milkrun

11,692.42 483.22 5.87 6.72 5.58 6.06 79.37 72.22 62.96 71.52

Outbound 2,709.00 398.97 6.72 - - 6.72 11.43 - - 11.43

Inbound 2,825.99 437.58 7.37 - - 7.37 6.35 - - 6.35

Total 5,534.99 836.55 14.08 - - 14.08 8.89 - - 8.89

Combined

Outbound and

Inbound Milkrun

2,958.00 481.14 8.1 - - 8.10 18.41 - - 18.41

Time (hrs) Utilization (%)

High

Demand

for pickp

and

delivery

Low

Demand

for pickp

and

delivery

Case

Table 5: Improvement after using combined inbound and outbound milk-run delivery

Total cost Total

distance

(baht) (km) Truck1 Truck2 Truck3 Average Truck1 Truck2 Truck3 Average

High

Demand

for pickp

and

delivery

Percent

Improvement38.6% 67.2% 6.8% -39.1% 58.8% 26.4% 81.8% 239.1% 15.9% 79.9%

Low

Demand

for pickp

and

delivery

Percent

Improvement87.1% 42.5% 42.5% - - 42.5% 107.1% - - 107.1%

Utilization (%)Case

Time (hrs)

4. CONCLUSION In most research, the milk-run delivery cannot be planned for both inbound and

outbound logistics due to the complexity of keeping inventory record on truck. When

allowed, the truck must complete all deliveries first before pick-up can be started. This

limits the opportunities to save transportation cost and increase truck capacity utilization.

We develop a mathematical model to allow planning of multiple trucks with combined

inbound and outbound milk-run deliveries. There is no restriction of delivery or pick-up

sequence imposed, thereby allowing a truck to travel freely to minimize the total cost. The

model is tested in a small numerical example with eight nodes. From the numerical result,

the solution from the model shows the savings of 38.6% and 87.1% for the high demand

and low demand cases respectively. The total distance and average travel time also

improve at least 42.5% and 26.4% respectively. The average truck capacity utilization is

also improved at least 79.9%. As the improvement is quite significant, the proposed

method shows a viable solution to save the logistics cost. In the future research, more

numerical results will be tested to show the gained benefits. Also, to be able to keep

tracks of inventory on truck, the time dimension is added to the variables. This increases

the complexity of solving the large size problems. The efficient heuristic method will be

developed in the future research.

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References

Dianwei, Q (2006), The Research on Logistics Cost Accounting and Management in

China, Proceedings of 2006 International Conference on Management of

Logistics and Supply Chain: September 20-22, 2006, Sydney, Australia

Hakan Yildiz, R., Ravi B, Wayne Fairey C. (2010) “Integrated optimization of customer

and supplier logistics at Robert Bosch LLC,” European Journal of Operational

Research, 207 (1), 456–464.

Hugos, M. H. (2006) Essentials of Supply Chain Management, chapter 6, 185–186.

Peetijade, C. and Bangviwat, A. (2012) Empty Trucks Run Reduction in Bangkok Area

Towards Sustainable Transportation, International Journal of Trade, Economics

and Finance, 3 (2), 91-95

Raju, P. N., Ajitkumar, B. S. and Dhake, S. (2007) Implementation of Milk Run Logistics

System in an Auto Component Manufacturing Plant. SASTECH Journal,

MSRSAS, 7, 27-32.

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